Notes of Enrico Le Donne’s lecture

Sub-Riemannian geometry working seminar, organizational matters

– We meet every wednesday, 5pm, at IHP, amphi Darboux (except for a few exceptional days where we move to room 05, same floor). If people prefer a different schedule, let me know.

– People welcome to join for dinner after seminar.

– Many people have offered to give a talk. After today’s talk, we shall sort out proposals.

1. Metric spaces not parametrized by their tangents

1.1. Sub-Finsler manifolds

Definition 1 An (equiregular) sub-Finsler manifold is the data of a Finsler manifold (a smoothly varying norm is given on each tangent space) and a smooth sub-bundle ${\Delta\subset TM}$ satisfying the following assumptions. Let ${\Xi^k}$ denote the family of vector fields obtained iteratively from ${\Xi^1 =}$ smooth sections of ${\Delta}$ by the Lie bracket operation,

$\displaystyle \begin{array}{rcl} \Xi^{k+1}=[\Xi^1,\Xi^k]. \end{array}$

Let ${\Delta^k =\Xi^k (p)}$ denote the set of values at point ${p}$ of elements of ${\Xi^k}$. Then we assume that

1. There exists ${s}$ such that ${\Delta^s =TM}$.
2. All ${\Delta^k}$ are sub-bundles.

A sub-Finsler manifold comes up with a distance: given two points, minimize length of curves between them which are almost everywhere tangent to ${\Delta}$. Note this distance only depends on the restriction of the norm to ${\Delta}$. So in fact one does not need the norm is other directions. It is a leitmotiv of sub-Finsler geometry: although, occasionnally, we may use geometric data defined outside ${\Delta}$, what really matters for us is ${\Delta}$ and the associated metric.

1.2. Carnot groups

Carnot groups form an important class of examples.

Definition 2 Let ${\mathfrak{g}}$ be a Lie algebra. Say that a direct sum decomposition ${\mathfrak{g}=V^1 \oplus\cdots\oplus V^s}$ is a stratification of ${\mathfrak{g}}$ if

1. ${[V^i ,V^j]\subset V^{i+j}}$.
2. ${V^{k+1}=[V^1 ,V^k]}$.

Pick a norm on ${\mathfrak{g}}$, left-translate it, this gives a sub-Finsler manifold.

Example 1 ${\mathfrak{g}=V^1 \oplus V^2}$ with ${V^1={\mathbb R}^2}$, ${V^2={\mathbb R}}$ and ${[\cdot,\cdot]:V^1 \times V^1 \rightarrow V^2}$ is the determinant. This is called the first Heisenberg Lie algebra.

Picture…

1.3. Tangent cones

Theorem 3 (Mitchell 1985, completed by other people) The tangent cone of a sub-Finsler manifold et a point ${p}$ is a Carnot group (which may depend on ${p}$).

Here, tangent cone of ${M}$ at ${p}$ means the limit of dilates of ${M}$. Limit is taken in Gromov-Hausdorff sense.

Definition 4 Let ${X}$, ${Y}$ be bounded metric spaces. Define

$\displaystyle dist_{GH}(X,Y)=\inf dist_{H}(X'\subset Z,Y'\subset Z),$

where inf is taken over all metric spaces ${Z}$, and all subsets ${X'\subset Z}$ isometric to ${X}$ and ${Y'\subset Z}$ isometric to ${Y}$, and maps preserve base points.

For unbounded pointed metrc spaces ${(X_n,x_n)}$ and ${(Y,y)}$, say that ${(X_n,x_n)}$ tends to ${(Y,y)}$ if for all ${R>0}$, the ${R}$-balls converge, i.e.

$\displaystyle \begin{array}{rcl} dist_{GH}(B^{X_n}(x_n,R),B^{Y}(y,R))\rightarrow 0. \end{array}$

Here we are in fact dealing with (pointed) metric spaces up to isometry.

Metric spaces need not admit tangent cones. The doubling condition is a favorable circumstance: it guarantees compactness of dilates, and so existence of converging subsequences among dilates. Nevertheless, limits need not exist (i.e. sub-limits need not be unique), even in the class of doubling metric spaces.

Theorem 5 (Le Donne) Let ${X}$ be a geodesic metric space. Assume ${X}$ has a doubling measure ${\mu}$. Assume that ${\mu}$-almost everywhere, ${X}$ has only one tangent cone (i.e. dilates converge to some metric space). Then the tangent cone is a Carnot group ${\mu}$-almost everywhere.

The proof uses the following

Lemma 6 In a doubling metric space ${X}$, almost every point ${x}$ has the following property. Assume ${(Y,y)}$ is a tangent cone to ${X}$ at ${x}$. Then for all ${y'\in Y}$, ${(Y,y')}$ is again a tangent cone of ${X}$ at ${x}$.

and relies essentially on the following

Theorem 7 (Gleason, Montgomery, Zippin 1950, Berestovskii 1988) Assume ${Y}$ is a complete, proper, connected and locally connected metric space. Assume isometry group is transitive. Then ${Isom(Y)}$ is a Lie group with finitely many connected components.

Moreover (Berestovskii), if ${Y}$ is geodesic, then ${Y}$ is a sub-Finsler manifold.

1.4. Impossible parametrization by tangents

For sub-Riemannian manifolds, the tangent space is not a local model in general.

Theorem 8 (Le Donne, Ottazzi, Warhurst 2011) There exists a nilpotent Lie group equipped with a left-invariant sub-Riemannian metric which is not locally quasiconformal to its tangent cones.

Note that bi-Lipschitz ${\Rightarrow}$ quasiconformal.

Definition 9 Say a Carnot group is ultra rigid if every even locally defined quasiconformal self-mapping is the restriction of the composition of a translation and a dilation.

Theorem 8 easily follows from the following

Lemma 10 There exists a nilpotent non Carnot Lie group ${H}$ equipped with a left-invariant sub-Riemannian distance for which the tangent cone ${G}$ is ultra rigid.

Indeed, ultra rigidity provides an injective group homomorphism from ${H}$ to the group ${G\times\mathbb{R}}$ of translations/dilations of ${G}$. The semi-direct product is not nilpotent, so homomorphism is not onto (at the level of Lie algebras), ${\mathfrak{h}}$ maps injectively to ${\mathfrak{g}}$. Since they have the same dimension, ${\mathfrak{h}=\mathfrak{g}}$, contradiction.